I wish to rotate a body that is located on $\hat z$ axis.
If I rotate the body at angle $\alpha$ around the $\hat x$ axis and then at angle $\beta$ around the $\hat y$ axis then I think I should get:
$$R_y(\beta)R_x(\alpha)=\begin{bmatrix}\cos(\beta) && 0 && \sin(\beta)\\ 0 && 1 && 0 \\ -\sin(\beta) && 0 && \cos(\beta)\end{bmatrix} \begin{bmatrix}1 && 0 && 0\\ 0 && \cos(\alpha) &&-\sin(\alpha) \\ 0 && \sin(\alpha) && \cos(\alpha) \end{bmatrix}$$
According to my TA this should give:
$$\begin{bmatrix} \cos(\beta) && \cos(\alpha)\sin(\beta) && \sin(\alpha)\sin(\beta) \\ 0 && -\sin(\alpha) && \cos(\alpha) \\ -\sin(\beta) && \cos(\alpha)\cos(\beta) && \cos(\beta)\sin(\alpha) \end{bmatrix}$$
but when I multiply it myself I get something else.
Either I don't know how to multiply matrices, I did not use the the right rotation matrices, or the TA is wrong.
Best Answer
The correct answer is$$\begin{bmatrix} \cos (\beta) & \sin (\alpha) \sin (\beta) & \cos (\alpha) \sin (\beta) \\ 0 & \cos (\alpha) & -\sin (\alpha) \\ -\sin (\beta) & \cos (\beta) \sin (\alpha) & \cos (\alpha) \cos (\beta) \end{bmatrix}.$$If this is what you got, then your TA is wrong.